Extended search for sub-eV axion-like resonances via four-wave mixing with a quasi-parallel laser collider in a high-quality vacuum system

Resonance states of axion-like particles were searched for via four-wave mixing by focusing two-color pulsed lasers into a quasi-vacuum. A quasi-parallel collision system that allows probing of the sub-eV mass range was realized by focusing the combined laser fields with an off-axis parabolic mirror. A 0.10 mJ/34 fs Ti:Sapphire laser pulse and a 0.14 mJ/9 ns Nd:YAG laser pulse were spatiotemporally synchronized by sharing a common optical axis and focused into the vacuum system. No significant four-wave mixing signal was observed at the vacuum pressure of $3.7 \times 10^{-5}$ Pa , thereby providing upper bounds on the coupling-mass relation by assuming exchanges of scalar and pseudoscalar fields at a 95 % confidence level in the mass range below 0.21 eV. For this search, the experimental setup was substantially upgraded so that optical components are compatible with the requirements of the high-quality vacuum system, hence enabling the pulse power to be increased. With the increased pulse power, a new kind of pressure-dependent background photons emerged in addition to the known atomic four-wave mixing process. This paper shows the pressure dependence of these background photons and how to handle them in the search.


I. INTRODUCTION
The spontaneous breaking of a global symmetry accompanies a massless Nambu-Goldstone boson (NGB) [1] . In nature, however, such an NGB possesses a finite mass due to complicated quantum corrections. The neutral pion is such a pseudo-NGB (pNGB) state, gaining mass because of chiral symmetry breaking in quantum chromodynamics (QCD). The concept of spontaneous symmetry breaking can be a robust guiding principle to understand dark matter and dark energy in the context of particle physics. Indeed, several theoretical models predict low-mass pNGBs such as the QCD axion [2] for solving the strong CP problem, and pNGBs in the contexts of string theory [3] and unified inflation and dark matter [4], which are commonly pseudoscalar fields. As a scalar-field example, dilaton is predicted as a source of dark energy [5]. These weakly coupling pNGBs are known generically as axion-like particles (ALPs). Because it is difficult to evaluate the physical masses of pNGBs theoretically, model-independent laboratory experiments are indispensable for determining the physical masses as comprehensively as possible at relatively low masses compared to those of high-energy charged-particle colliders. Because a pNGB is a low-mass state and in principle unstable even if the lifetime is long, the resonance state can be produced directly by colliding massless particles such as photons as long as the pNGB has coupling to photons. In previous works, we have reported on the first search for scalar-type pNGBs with laser beams in a quasi-parallel collision system (QPS) [6], and then the second search for sub-eV scalar and pseudoscalar fields in the QPS [7].
A QPS can be realized conceptually by focusing a laser beam with a focusing optical component as illustrated in Fig. 1. The solid (green) laser field with photon energy ω is focused, and the center of mass system (CMS) energy between a randomly selected photon pair within the focused field is expressed as where ϑ is half of the relative incident angle of the photon pair. By adjusting the beam diameter d and focal length f , we can control the range of possible incident angles within ∆θ, that is, the accessible range of E CM S in a single focusing geometry. Because ϑ can be close to zero, the QPS is in principle sensitive to pNGBs with almost zero mass. Consequently, the photon-photon scattering in the QPS is accessible to the mass range 0 < m < 2ω∆θ, although the sensitivity to a lower mass range is steeply suppressed. Capturing a resonant pNGB within an E CM S uncertainty via an s-channel exchange is the first key element of this method to increase the interaction rate. The second key element is in stimulation to guide the scattering to a fixed final state. The dashed (magenta) laser beam with photon energy uω (0 < u < 1) indicates the inducing laser field to stimulate the interaction, where the probability of generating a signal photon with energy (2 − u)ω is enhanced via energymomentum conservation in ω + ω → pNGB → uω + (2 − u)ω by the coherent nature of the co-propagating inducing laser. The scattering probability increases in proportion to (i) the number of photons in the inducing laser and (ii) the square of the number of photons in the creation laser [8][9][10][11]. This process is kinematically similar to four-wave mixing in the context of atomic physics [12].
Herein, we report the results of the third search by increasing the laser intensities with the upgraded experimental systems. To increase the creation laser intensity in particular, we used a Ti:Sapphire-based laser, T 6 -laser at the Institute for Chemical Research at Kyoto University, which can produce a 10 TW pulse intensity at maximum peak power at a repetition rate of 5 Hz. For this upgrade, main optical components were installed in the vacuum system which consists of transport chambers and an interaction chamber separately. We mostly respected construction of the high-quality vacuum system, especially for the interaction part, which required downsizing of the optical paths and also the capabil-ity to manipulate optical components from outside these chambers. The main purpose of the present search is therefore to show the extensibility of this searching method in such a high-quality vacuum system, even if new types of background sources emerge as a result of increased laser intensities.

II. COUPLING-MASS RELATION CONFIGURED FOR THE SEARCH
In order to discuss the coupling of scalar (φ) or pseudoscalar (σ) fields to two photons, we introduce the following two effective Lagrangians: where F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor and its dualF µν is defined as ǫ µναβ F αβ with the Levi-Civita symbol ǫ ijkl , and g is a dimensionless constant while M is a typical energy at which a relevant global symmetry is broken.
The scalar and pseudoscalar fields couple to the two photons differently depending on the linear polarization states of the two photons. In the photon-photon scattering process p 1 + p 2 → p 3 + p 4 in four-momentum space, when all photons are on an identical reaction plane, that is, in the case of the coplanar condition, the distinction between scalar and pseudoscalar field exchanges becomes prominent as follows. Denoting individual linear polarization states in parentheses, the non-vanishing scattering amplitudes are limited to for the scalar field exchange and for the pseudoscalar field exchange, where swaps between (1) and (2) give the same scattering amplitudes, respectively.
While this coplanar condition is always satisfied in CMS, it is not true in the case of QPS as illustrated in Fig.2, where the p 1 − p 2 plane and the p 3 − p 4 plane may differ from the x L − z L plane in the laboratory coordinates. We define experimental linear polarization Definitions of linear polarization directions and rotation angles in focused collision geometry. This figure is reproduced from [7] with a slight modification.
states {1} and {2} by mapping them along y L and x L axes, respectively. We then define rotation angles of these reaction planes as Φ and ϕ for initial and final state photon pairs, respectively, with respected to the x L − z L plane.
In this search, indeed, we will assign the P-polarized state of the creation laser to {1} state and the S-polarized state of the inducing laser to {2} state, because the orthogonal combination of P-and S-polarization states is very effective to suppress the known atomic four-wave mixing process in the residual gas [7,12].
For a given set of fixed laboratory polarization directions {1} and {2}, we have introduced an axially asymmetric factor F abcd with linear polarization states ab and cd for the initial and final state photon pairs, respectively, by fixing Φ = 0 for scalar and pseudoscalar cases, respectively [9]. If there is no inducing field in the final state, p 3 − p 4 planes rotate around the z L axis symmetrically resulting in the axial symmetric factor of 2π due to the solid angle integration of signal photons. However, since we induce p 3 photons as the signal by supplying the polarization-fixed coherent p 4 field in the search, the p 3 − p 4 plane rotation factor deviates from 2π depending on types of exchanged fields. So the F -factor corresponds to the replacement of the symmetric solid angle integral.
In addition we have further introduced an incident plane rotation factor G ab in QPS [7], because theoretically introduced p 1 − p 2 planes are independent of the experimentally introduced x L − z L plane based on {1} and {2} states of creation and inducing laser fields, respectively. Thanks to this situation, even if the polarization direction of the creation laser field is fixed at {1} state in the laboratory coordinates, the searching system can also be sensitive to the pseudoscalar field exchange through the rotation of the p 1 − p 2 plane. The G-factor corresponds to an averaged correction factor with respect to the case with Φ = 0 by taking possible rotation angles over Φ = 0 ∼ 2π.
In this search we thus focus on the following scattering processes in the same way as the previous publication [7]: with the assumption of the scalar field exchange and with that of the pseudoscalar field exchange.
For given parameters for the pulsed lasers and optical components, the coupling strength g/M is related to the yield Y based on Eq. (26) in Appendix herein, namely, the number of where the subscripts c and i denote the creation and inducing lasers, respectively. This relation is almost the same as the one used for the second search [7] except for the condition that the beam diameters of the creation and inducing lasers are non-negligibly different in the present search, so the common diameter parameter for both beams cannot be assigned.
The relevant modifications are summarized in Appendix herein, while all the other details are given in the appendices of the previous searches [6,7]. The individual parameters are then summarized as follows: ω c is the incident photon energy of the creation laser, λ c,i are the wavelengths of the two lasers, τ c,i are the pulse durations, f is the common focal length, With the central wavelengths λ c and λ i for the creation and inducing lasers, respectively, the central wavelength λ s of the signal is defined via energy conservation as with ∼ 40 ps time-resolution to monitor the stability of the pulse energies and to adjust the time synchronization between the creation and inducing laser pulses.
As a signal detector, we used an R7400-01 single-photon-countable photomultiplier tube (PMT) manufactured by HAMAMATSU which has a falling time resolution of 0.75 ns.
In order for the detection system to be sensitive only to signal photons between 610 and 690 nm, the PMT was installed after an LPF transmitting above 610 nm and three types of band-pass filters transmitting 570-800 nm, 500-930 nm, and 450-690 nm to remove residual photons from the intense incident lasers We acquired waveform data of analog currents from the PMT and the two PDs using a waveform digitizer triggered with a basic 10 Hz trigger to which both the creation and inducing laser injections were synchronized as shown in Fig. 5. Two mechanical shutters reduced the injection rate of the creation laser down to equi-interval 5 Hz and that of the inducing laser to non-equi-interval 5 Hz for waveform analysis as indicated in Fig. 5. The two beams were injected with different timing patterns so that the waveform data could be recorded with the following four consecutive trigger patterns with equal statistics: (i) both beams were incident "S", (ii) only the inducing laser was incident "I", (iii) only the creation laser was incident "C", and (iv) neither beam was incident "P".

IV. MEASUREMENT OF FOUR-WAVE MIXING PROCESS IN RESIDUAL GAS
The observed number of photons was estimated by analyzing the waveform data from the PMT. An individual waveform consisted of 1000 sampling data points within a 500 ns time window. Figure 6 shows an example of waveform data in which a peak is identified. Negative peaks were identified with a peak finder applying the same algorithm as that in the previous search [7]. We then calculated the charge sums in the peak structures. The charge sums in the peak structures were converted to the number of photons by dividing by the measured single-photon equivalent charge. Four-wave mixing photons are expected to be caused by residual gases in the interaction chamber as a background source. However, this photon source can be used as a calibration source to assure the spatiotemporal synchronization of the creation and inducing laser pulses.
To quantify the number of background photons, we measured the pressure dependence of the number of four-wave mixing photons when the linear polarization directions of the creation and inducing lasers were parallel to enhance the atomic four-wave mixing process. Figure 7 shows The number of four-wave mixing photons, N S , was then evaluated from where n T is the number of photons for a trigger pattern T entering the time window 262.5-264 ns, which is consistent with the signal generation timing. Figure 8 shows the pressure dependence of the number of signal photons per shot, which was fitted with where subscripts c and i indicate the creation and inducing lasers, respectively and N(x, y) is the local intensity per CCD pixel of the monitor camera. The summations were over the area framed by FWHM of the creation laser intensity profile. Fluctuations of the overlap factors were then estimated as where O I,F are the overlap factors at the beginning and end, respectively, of a unit run period during 1200 s, and deviations with respect to the mean overlap factor (O I + O F )/2 were calculated. The value of b = 1.82 ± 0.14 from the fitting results is close to the expected behavior N S ∝ P 2 in atomic four-wave mixing processes [7,12]. A notable difference from our previous publication [7] was that signal-like photons were indeed produced even in trigger pattern C. Figure 9 shows the number of photons, n c − n p , in trigger pattern C within the signal timing window after subtracting the side-band background photons from the peak part as a function of pressure. This indicates that n c − n p scales with P 2.2 and reaches zero within the statistical uncertainty below 1 Pa.
Because n c − n p is strongly pressure dependent, we conclude that this background source is generated from IP but different from the conventional four-wave mixing process because it does not require the inducing laser to produce the same frequency as that of four-wave mixing. Creation of a plasma state at IP by the higher-power creation laser might explain the broad-band emissions, though this is not directly relevant to the present search because we can subtract trigger pattern C from pattern S.

V. SEARCH FOR FOUR-WAVE MIXING SIGNALS IN VACUUM
We searched for scalar-and pseudoscalar-type resonance states at a vacuum pressure of 3.7 × 10 −5 Pa by combining P-pol. for the creation pulsed laser and S-pol. for the inducing pulsed laser.
When the polarization directions of the two beams were orthogonal, P-pol.(creation) + S-pol.(inducing), counting the number of four-wave mixing photons in the similar pressure range to that of the P-pol.(creation) + P-pol.(inducing) case was rather difficult because the number of photons in trigger pattern C was considerable and the subtraction from trigger pattern S was dominated by the statistical uncertainty. However, we know that the parallel case dominates the orthogonal case at a higher pressure range above 10 2 Pa [7]. Thus, the pressure dependence in Fig. 8 can give an upper limit on the number of photons from the atomic four-wave mixing process even without the measured pressure dependence for the orthogonal case, which is the actual searching configuration as discussed in Section II.
where the upper limit was estimated by taking the maximum value based on the fitting error on the parameter b. Therefore, in the case of the orthogonal polarization combination, the expected number of background photons caused by the residual atoms is below unity as long as the total statistic is below 10 11 shots. The absolute values of laser pulse energies for the creation and inducing lasers were 101±4 µJ and 300 ± 5 µJ, respectively, at the beginning of the first run, while they were 98 ± 4 µJ and 303 ± 5 µJ, respectively, at the end of the final run. So the absolute pulse energies of the two lasers were quite stable over the entire data taking period for 40 hours. This overlap factor reflects the nature of four-wave mixing on the cubic intensity dependence.
Systematic error III is an error caused by the threshold value used for the peak finding.
This was calculated as two standard deviation assuming that the number of photon-like signals obtained by varying V threshold from −1.2 to −1.4 mV follows a uniform distribution. No statistically significant four-wave mixing photons in the quasi-vacuum state were observed in this search from the result in (14). We thus set the exclusion regions on the coupling-mass relation by assuming scalar and pseudoscalar fields based on the parameter values summarized in Tab.I.
First, the upper limit on the sensitive mass range is estimated as based on values in Tab.I, where ∆θ is defined by the focal length f and beam diameter d of the creation laser in Fig. 1 and ϑ varies from zero to ∆θ.
The efficiency-corrected number of four-wave mixing photons, N S , was evaluated from the following relation with the relevant experimental parameters: and G ps 12 are averaging factors by the incident reaction plane rotation for scalar and pseudoscalar field exchanges, respectively. The evaluation of G is discussed in the appendix of the previous search [7]. F sc 1122 and F ps 1212 are axially asymmetric factors for scalar and pseudoscalar field exchanges, respectively, as explained in the appendix of [9]. Note that the effective N i is deduced from the overlap between the measured focal images of the inducing and creation lasers within the area framed by ±3 standard deviation in the Gaussian-like intensity profile of the creation laser.  where ǫ opt is the acceptance factor for signal photons to propagate from IP through several optical components, and ǫ D is the detection efficiency of the PMT due mainly to the quantum efficiency of the photocathode when a single photon enters. ǫ opt includes the inefficiencies from IP down to the actual location of the PMT, namely, those of OAP2, DM3-DM7, W5, L1-L2, and the transmittance of the 16 wave filters in front of the PMT. This factor was evaluated by taking the ratio between the calibration beam energy at the focal point and that at the detection point. The energy ratio was measured based on the intensity distributions at the common camera. In advance of the search, ǫ D was measured with 532 nm laser pulses.
We corrected the difference of the quantum efficiencies between 532 nm and 651 nm based on the relative quantum efficiencies provided by the HAMAMATSU specification sheet.
The upper limits on the coupling-mass relation at a 95 % confidence level were obtained based on the null hypothesis that the fluctuations of the number of signal photons follow Gaussian distributions whose expectation value, µ, is zero for the given total number of shots, W S = 4.2 × 10 4 . This null hypothesis is justified. First, the expectation value based on the unique standard model process, that is, the QED photon-photon scattering process is negligibly small at E cms < 1 eV [13] even if we add the stimulation effect [14] for the given shot statistics. Second, as discussed with inequality Eq. (13), the expectation value due to atomic processes in the residual gas is well below unity. Therefore, the background source is limited to only noises in the experimental environment. The systematic error I is the dominant component of the systematic errors. This was estimated by measuring the rootmean-square of the number of photon-like signals excluding the signal window. In fact, the measured N s distribution excluding the signal time window follows almost a perfect Gaussian distribution around its central value, though the central value itself has no specific meaning with respect to the signal timing window. Since the distribution is symmetric around the central value, a Gaussian distribution can be supported from the measurement whatever the physics behind it is. The Gaussian distribution is also deduced from the principle point of view. What we measure is accidentally incident photon-like waveforms in arbitrary time window (e.g. thermal noises, ambient photons from lasers and not from lasers, electric noises, and so forth) which should follow binomial distributions if the incident probabilities are individually constant over the measuring time. Although the incident probabilities are quite low, the expectation values of background photon-like signals per time bin over the data acquisition time are around 100 as seen in trigger pattern S and C in Fig.11. Therefore, we may expect that the numbers of background photons per time bin can be approximated as Gaussian distributions because the expectation value is large enough. N s then eventually corresponds to subtractions between these Gaussian distributions. We thus assume that the Gaussian distribution is the most natural null hypothesis in our search. In order to set a confidence level 1 − α to exclude the null hypothesis, we assign the acceptance-uncorrected uncertainty δN S which was evaluated as the quadratic sum of statistical and systematic errors in Eq. (14) as the one standard deviation σ to the following Gaussian kernel: where µ = 0 and N S corresponds to the estimator x in our case. In order to give a confidence level of 95 % in this analysis, we apply 2α = 0.05 with δ = 2.24σ where we set a one-sided upper limit by excluding above x+δ [15]. The upper limit of the signal yield to be compared to the theoretical calculation, namely, Y, was then evaluated as Based on the coupling-mass relations in Eq. (7), Figs. 13 and 14 show the obtained exclusion limits for scalar and pseudoscalar fields, respectively, at a 95 % confidence level.

VII. CONCLUSIONS
We performed an extended search for scalar and pseudoscalar fields via four-wave mixing by focusing two-color pulsed lasers, namely, 0.10 mJ/34 fs at 808 nm and 0.14 mJ/9 ns at 1064 nm. The observed number of four-wave mixing photons in the quasi-vacuum state at 3.7×10 −5 Pa was 5.4±30.7(stat.)±25.1(syst.I)±10.1(syst.II)±0.9(syst.III). We thus conclude that no significant four-wave mixing signal was observed in this search. The expected number of four-wave mixing photons from the residual gas is sufficiently small based on the upper limit from the measurement of the pressure dependence. With respect to an assumption that uncertainties are dominated by systematic fluctuations around the zero expectation value following the Gaussian distribution, we provided the upper limits on the couplingmass relations for scalar and pseudoscalar fields at a 95 % confidence level in the mass range below 0.21 eV, respectively. The upper limits on the coupling at m = 0.21 eV were obtained as 4.6 × 10 −6 GeV −1 and 5.1 × 10 −6 GeV −1 for scalar and pseudoscalar fields, respectively.

VIII. DISCUSSION AND FUTURE PROSPECT
For this extended search, we upgraded the searching system so that it can store multiple dichroic mirrors to select feeble signal photons among a huge number of background laser fields co-linearly propagating with the signal photons in the high-quality vacuum system.
We prioritized construction of the high-quality vacuum system over rapid increase of the creation laser intensity, which is designed to achieve 10 −8 Pa if a metal shield is installed for the lid contact, by introducing the entrance window (W2 in Fig. 3) in the interaction chamber to completely decouple the upstream transport vacuum system where the pressure is much higher. However, this window material indeed became an obstacle against drastically increasing the laser intensity, because high-intensity laser fields must penetrate the window through which a new background source was created in addition to the residual atoms around IP. Therefore, the intensity improvement for the creation laser from the previously published search [7] was rather moderate in the present search. In this search, however, we have shown clearly that the null result can be obtained with the current searching method, and we succeeded in measuring the pressure scaling of the atomic four-wave mixing process and also another plasma-like phenomenon down to ∼ 1 Pa. For the next upgraded search, we thus plan to eliminate this window. Instead, a differential pumping system will be inserted between the transport vacuum system and the interaction chamber. Therefore, we may expect that we can drastically improve the sensitivity with ∼ 10 3 times higher creation laser intensity available at the T 6 -laser system in the upgraded system.

APPENDIX: MODIFICATIONS OF THE COUPLING-MASS RELATION DUE TO THE BEAM DIAMETER DIFFERENCE
We basically use the coupling-mass relation that was used for the previous searches [6,7], where a common beam diameter was assumed for both the creation (c) and inducing (i) laser beams. However, here we slightly modify the parametrization by taking different beam diameters d c and d i into account. With respect to equations for relevant parameter used in the Appendix of [6], we put superscript * for the modified parameter notations as follows.
For the density factor for the creation beam, namely, D c in Eq. (A29), we have